The m-Steiner Traveling Salesman Problem with online edge blockages

Abstract

We consider the online multiple Steiner Traveling Salesman Problem based on the background of the delivery of packages in an urban traffic network. In this problem, given an edge-weighted undirected graph $$G = (V, E)$$ , a subset $$D\subset V$$ of customer vertices, and m salesmen. For each edge in E, the weight w(e) is associated with the traversal time or the cost of the edge. The aim is to find m closed tours that visit each vertex of D at least once. We formulate the traffic congestion with k non-recoverable blocked edges revealed to the salesmen in real-time, meaning that the salesmen know about a blocked edge whenever it occurs. For the version to minimize the maximum cost of m salesmen (minmax mSTSP), we prove a lower bound and propose the ForestTraversal algorithm. The corresponding competitive ratio is proved to be linear with k. For the version to minimize the total cost of m salesmen (minsum mSTSP), we also propose a lower bound and the Retrace algorithm, where the competitive ratio of the algorithm is proved to be linear with k.